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Telescoping method, summation formulas, and inversion pairs

  • Based on Gosper's algorithm, we present an approach to the telescoping of general sequences. Along this approach, we propose a summation formula and a bibasic extension of Ma's inversion formula. From the formulas, we are able to derive several hypergeometric and elliptic hypergeometric identities.

    Citation: Qing-Hu Hou, Yarong Wei. Telescoping method, summation formulas, and inversion pairs[J]. Electronic Research Archive, 2021, 29(4): 2657-2671. doi: 10.3934/era.2021007

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  • Based on Gosper's algorithm, we present an approach to the telescoping of general sequences. Along this approach, we propose a summation formula and a bibasic extension of Ma's inversion formula. From the formulas, we are able to derive several hypergeometric and elliptic hypergeometric identities.



    The telescoping method aims to solve the following problem: For a sequence f(k) in some class S, decide whether there exists g(k)S such that

    f(k)=g(k+1)g(k).

    The sequence g(k) is called the anti-difference of f(k). Clearly, once we obtain g(k), we will derive the indefinite summation formula

    b1k=af(k)=g(b)g(a). (1)

    Gosper [7] solved the telescoping problem for hypergeometric terms by giving the so-called Gosper's algorithm. As mentioned in the book of Petkovše et al. [12], Gosper's algorithm is one of the landmarks in the history of symbolic summation. It not only fully solves the telescoping problem of hypergeometric terms, but also plays an important role in the creative telescoping algorithm developed by Wilf and Zeilberger [15].

    Since the appearance of Gosper's algorithm and Zeilberger's algorithm, the telescoping method for more general sequences has been extensively studied. Chyzak [4] extended Zeilberger's algorithm to holonomic sequences. Inspired by Karr's summation algorithm [8], Schneider [13] presented an approach to summation in difference rings. Recently, Paule and Schneider [11] gave a symbolic summation theory for unspecified sequences.

    This paper is motivated by considering the inversion formulas of basic and elliptic hypergeometric series. We find that the idea of Gosper's algorithm is an efficient mechanism for this purpose.

    Let us recall the main steps of Gosper's algorithm. Suppose tk is a hypergeometric term, i.e., the ratio tk+1/tk is a rational function of k. We first write the ratio tk+1/tk in the form

    tk+1tk=a(k)b(k)c(k+1)c(k), (2)

    where a(k),b(k),c(k) are polynomials in k such that gcd(a(k),b(k+h))=1 for all nonnegative integers h. Then a hypergeometric telescoping zk of tk exists if and only if there is a polynomial solution x(k) to the linear difference equation

    a(k)x(k+1)b(k1)x(k)=c(k). (3)

    If x(k) exists, the telescoping of tk is

    zk=b(k1)x(k)c(k)tk. (4)

    To deal with sequences which are not hypergeometric, we release the restriction that tk+1/tk is a rational function of k so that the a(k),b(k),c(k) in (2) are arbitrary functions of k. It is straightforward to check that if x(k) is a function satisfying (3), then the function zk given by (4) is still a telescoping of tk.

    This observation motivates a new approach to the definite summation of sequences beyond the hypergeometric ones. Given a sequence tk, we get (2) by spreading out the common factors of the numerator and the denominator of the ratio tk+1/tk up to a shift. Then we try to find a function x(k) satisfying (3). Once we have (2) and (3), we will obtain the telescoping zk and thus derive the summation formula (1).

    Recall that two upper triangle matrices (fn,k)n,k0,(gn,k)n,k0 are called an inversion pair if

    nk=0fn,kgk,l=δn,l={1,if n=l,0,otherwise. (5)

    If (f,g) is an inversion pair, we have

    nk=0fn,kF(k)=G(n)nk=0gn,kG(k)=F(n).

    There are many interesting applications of the inversion pairs. For example, Warnaar [14] established some new identities on multibasic theta hypergeometric series by elliptic analogue of the inverse relations. Ma gave an extension of Warnaar's matrix inversion and obtained a series of classical identities in [10].

    Ma [10] showed that the inversion relations (5) can be derived by summation formulas of form (1). This enables us to study inversion pairs by solving the telescoping problem. We will see that the trivial case of x(k)=1 leads to the key summation formula (Theorem 4) in [10]. Note that the assumption on f,g in Ma's formula (Equaiton (6) below) is nothing but the Gosper equation (3).

    Along this approach, we construct a new summation formula which extends Ma's result. Based on it and the related inversion pairs, we are able to derive several basic hypergeometric identities and elliptic hypergeometric identities.

    The paper is organized as follows. In Section 2, we give a simple proof of Ma's summation formula based on the telescoping method. Then in Section 3, we propose a bibasic extension of Ma's summation formula along the telescoping approach and give some applications. Then in Section 4, we present a summation formula corresponding to the non trivial case where x(k)1. Section 5 provides the inversion formulas related to the summation formulas.

    Notation. Throughout this paper, we will employ the following standard notations for the theory of q-series and elliptic functions [6]. Assume |q|<1, we define the q-shifted factorials by

    (a;q)=k=0(1aqk),and(a;q)n=(a;q)(aqn;q).

    Write

    (a1,a2,,am;q)n=mj=1(aj;q)n,

    an basic hypergeometric series rϕs is defined by

    rϕs[a1,a2,,arb1,,br;q,z]=k=0(a1,,ar;q)k(q,b1,bs;q)k[(1)kq(k2)]1+srzk.

    and the balanced basic hypergeometric series r+1ϕr is defined by z=q and the parameters satisfy b1b2br=qa1a2ar+1.

    Assume further that |p|<1. The elliptic function is defined by

    θ(x)=θ(x;p)=(x;p)(p/x;p).

    The elliptic analogues of the q-shifed factorials are given by

    (a;q,p)=k=0θ(aqk;p),and(a;q,p)n=(a;q,p)(aqn;q,p).

    Also, we write

    (a1,a2,,am;q,p)n=mj=1(aj;q,p)n,and(a;q,p)kn=(a,aq,,aqk1;qk,p)n.

    Note that θ(x;0)=1x and hence (a;q,0)n=(a;q)n.

    The balanced, very-well-poised, elliptic (or modular) hypergeometric series is defined by

    r+1ωr(a1;a4,,ar+1;q,p)=k=0θ(a1q2k;p)θ(a1;p)(a1,a4,,ar+1;q,p)k(q,a1q/a4,,a1q/ar+1;q,p)kqk,

    where the parameters satisfy (a4ar+1)2=ar31qr5.

    We follow the usual convention in defining produces as

    mj=kAj:={AkAk+1Am,mk,1,m=k1,(Am+1Am+2Ak1)1,mk2.

    Ma used the following summation formula to prove the inversion formulas. We will give a simple proof by the idea of Gosper's algorithm.

    Theorem 2.1. ([10,Theorem 4]). Let f(a,b) and g(a,b) be two arbitrary nonzero functions over C in variables a,b such that

    f(a,c)g(b,d)f(a,d)g(b,c)=f(a,b)g(c,d), (6)

    and ai,bi,ci,di be arbitrary sequences such that none of the denominators in (7) vanishes. Then for any nonnegative integers m,n,

    mk=nf(ak,bk)g(ck,dk)k1j=1f(aj,cj)g(bj,dj)kj=1f(aj,dj)g(bj,cj)=mj=1f(aj,cj)g(bj,dj)mj=1f(aj,dj)g(bj,cj)0j=nf(aj,dj)g(bj,cj)0j=nf(aj,cj)g(bj,dj). (7)

    Proof. Let

    tk=f(ak,bk)g(ck,dk)k1j=1f(aj,cj)g(bj,dj)kj=1f(aj,dj)g(bj,cj).

    We have

    tk+1tk=f(ak,ck)g(bk,dk)f(ak+1,dk+1)g(bk+1,ck+1)f(ak+1,bk+1)g(ck+1,dk+1)f(ak,bk)g(ck,dk).

    Set

    a(k)=f(ak,ck)g(bk,dk), b(k1)=f(ak,dk)g(bk,ck), c(k)=f(ak,bk)g(ck,dk).

    Then by the assumption (6), we have

    a(k)b(k1)=c(k).

    Therefore, the telescoping of tk is

    zk=b(k1)c(k)tk=k1j=1f(aj,cj)g(bj,dj)k1j=1f(aj,dj)g(bj,cj).

    we get the formula (7) by summing k from n to m.

    Motivated by Ma's summation formula, we consider another special kind of f(a,b) and derive a new summation formula.

    Theorem 3.1. Let f(a,b) be an arbitrary nonzero functions over C in variables a,b such that

    f(a,b)f(a,c)f(a,d)f(a,e)f(b,1)f(c,1)f(d,1)f(e,1)=bf(a,1)f(a,bc)f(a,bd)f(a,be), (8)

    and ai,bi,ci,di,ei be arbitrary sequences such that none of the denominators in (9) vanishes. Then for any nonnegative integers m,n, we have

    mk=nbkf(ak,1)f(ak,bkck)f(ak,bkdk)f(ak,bkek)×k1j=1f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)kj=1f(bj,1)f(cj,1)f(dj,1)f(ej,1)=mj=1f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)mj=1f(bj,1)f(cj,1)f(dj,1)f(ej,1)
    n1j=1f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)n1j=1f(bj,1)f(cj,1)f(dj,1)f(ej,1). (9)

    Proof. Let

    tk=bkf(ak,1)f(ak,bkck)f(ak,bkdk)f(ak,bkek)×k1j=1f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)kj=1f(bj,1)f(cj,1)f(dj,1)f(ej,1).

    Then,

    tk+1tk=bk+1f(ak+1,1)f(ak+1,bk+1ck+1)f(ak+1,bk+1dk+1)f(ak+1,bk+1ek+1)bkf(ak,1)f(ak,bkck)f(ak,bkdk)f(ak,bkek)×f(ak,bk)f(ak,ck)f(ak,dk)f(ak,ek)f(bk+1,1)f(ck+1,1)f(dk+1,1)f(ek+1,1).

    Set

    a(k)=f(ak,bk)f(ak,ck)f(ak,dk)f(ak,ek),b(k)=f(bk+1,1)f(ck+1,1)f(dk+1,1)f(ek+1,1),c(k)=bkf(ak,1)f(ak,bkck)f(ak,bkdk)f(ak,bkek).

    We see that (2) holds. By the assumption (8), we have

    a(k)b(k1)=c(k).

    Hence by the Gosper's algorithm, we obtain the telescoping of tk:

    zk=b(k1)c(k)tk=k1j=1f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)k1j=1f(bj,1)f(cj,1)f(dj,1)f(ej,1).

    we get the formula (9) by summing k from n to m, .

    Now we give some applications of Theorem 3.1. The first one is a summation formula for finite series.

    Corollary 3.1. Let ai,bi,ci,di,ei be arbitrary sequences with a2i=bicidiei such that none of the denominators in (10) vanishes. Then for any nonnegative integer m, we have

    mk=nbk(1ak)(1ak/bkck)(1ak/bkdk)(1ak/bkek)×k1j=1(1aj/bj)(1aj/cj)(1aj/dj)(1aj/ej)kj=1(1bj)(1cj)(1dj)(1ej)=mj=1(1aj/bj)(1aj/cj)(1aj/dj)(1aj/ej)mj=1(1bj)(1cj)(1dj)(1ej)n1j=1(1aj/bj)(1aj/cj)(1aj/dj)(1aj/ej)n1j=1(1bj)(1cj)(1dj)(1ej). (10)

    Proof. Let f(a,b)=1a/b. It is straightfoward to check that (8) holds by the assumption a2i=bicidiei. Hence by Theorem 3.1, we derive (10).

    Corollary 3.1 is equivalent to a result of Ian G. Macdonald, first published by Bhatnagar and Milne [2,Eq.(2.30)]. From this formula, we can derive several summation identities. Here are two examples.

    Example 3.1. In Corollary 3.1, set

    ak=aq2k, bk=qk, ck=aqk/b, dk=aqk/c, ek=aqk/d,

    where a=bcd. We derive that

    mk=0(a,qa1/2,qa1/2,b,c,d;q)k(q,a1/2,a1/2,aq/b,aq/c,aq/d;q)k(aqbcd)k=(aq,aq/bc,aq/bd,aq/cd;q)m(aq/b,aq/c,aq/d,aq/bcd;q)m.

    Let m, we obtain a balanced 6ϕ5 series [6,Page 356,Eq. (Ⅱ.20)]

    _6ϕ5[a,qa1/2,qa1/2,b,c,da1/2,a1/2,aq/b,aq/c,aq/d;q,aqbcd]=(aq,aq/bc,aq/bd,aq/cd;q)(aq/b,aq/c,aq/d,aq/bcd;q).

    Example 3.2. Setting

    ak=adqkpk, bk=dqk, ck=adqk/b, dk=adpk/x, ek=bxpk/d

    in Corollary 3.1, we derive an indefinite bibasic summation formula [6,Eq. (Ⅱ.36)]:

    mk=n(1adpkqk)(1bpkqk/d)(1ad)(1b/d)(a,b;p)k(x,ad2/bx;q)k(dq,adq/b;q)k(adp/x,bpx/d;p)kqk=(1a)(1b)(1x)(1ad2/bx)(1ad)(1b/d)(dx)(1ad/bx){(ap,bp;p)m(xq,ad2q/bx;q)m(dq,adq/b;q)m(adp/x,bpx/d;p)m(1/d,b/ad;q)n+1(x/ad,d/bx;p)n+1(1/a,1/b;p)n+1(1/x,bx/ad2;q)n+1}.

    Gasper and Rahman used this identity to set up a series of quadratic and cubic summation and transformation formulas of basic hypergeometric series.

    The second one is an elliptic hypergeometric identity.

    Corollary 3.2. Let ai,bi,ci,di,ei be arbitrary sequences with a2i=bicidiei such that none of the denominators in (11) vanishes. Then for any nonnegative integer m,n, we have

    mk=nbkθ(ak,ak/bkck,ak/bkdk,ak/bkek;p)k1j=1θ(aj/bj,aj/cj,aj/dj,aj/ej;p)kj=1θ(bj,cj,dj,ej;p)=mj=1θ(aj/bj,aj/cj,aj/dj,aj/ej;p)mj=1θ(bj,cj,dj,ej;p)n1j=1θ(aj/bj,aj/cj,aj/dj,aj/ej;p)n1j=1θ(bj,cj,dj,ej;p). (11)

    Proof. Let f(a,b)=θ(a/b;p). Noting that a2=bcde, we have Weierstrass' identity

    θ(a/b,a/c,a/d,a/e;p)θ(b,c,d,e;p)=bθ(a,a/bc,a/bd,a/be;p),

    which means that f(a,b) satisfies (8). Hence by Theorem 3.1, we derive (11).

    This formula has been stated in an equivalent form by Warnaar [14,Eq.(3.2)]. We list some applications of Corollary 3.2 on elliptic hypergeometric series.

    Example 3.3. Setting

    ak=q2ka, bk=qk, ck=qka/b, dk=qka/c, ek=qka/d

    and n=0 in Corollary 3.2 where a=bcd, and making some simplifications, we deduce that a special case of [6,Page 321,Eq.(11.4.4)] where a=bcd

    m+1k=0θ(q2ka;p)θ(a;p)(a,b,c,d,a2qm+2/bcd,qm1;q,p)k(q,qa/b,qa/c,qa/d,bcdqm1/a,aqm+2;q,p)kqk=(qa,qa/bc,qa/bd,qa/cd;q,p)m+1(qa/b,qa/c,qa/d,qa/bcd;q,p)m+1.

    Example 3.4. Setting

    ak=aqkrk, bk=qk/bd, ck=adqk, dk=crk, ek=abrk/c

    and n=0 in Corollary 3.2, we get

    mk=0qkθ(aqkrk,bqkrk;p)θ(a,b;p)(abd,1/d;r,p)k(cr,abr/c;r,p)k(a/c,c/b;q,p)k(q/bd,adq;q,p)k=θ(c,ab/c,bd,ad;p)θ(a,b,abd/c,cd;p)(1(abd,drm;r,p)m+1(rm/c,ab/c)m+1(a/c,bqm/c;q,p)m+1(bdqm,ad;q,p)m+1).

    It is an indefinite elliptic hypergeometric series of Warnaar [14]. Moreover, Warnaar used the specialization of this identity to set up a pair of inverse matrices.

    Example 3.5. Setting

    ak=ad(rst/q)k, bk=dqk, ck=adb(st/q)k, dk=adc(rt/q)k, ek=bcd(rs/q)k

    in Corollary 3.2, and making some simplifications, we deduce that [6,Page 326,Eq.(11.6.6)]

    nk=mθ(ad(rst/q)k,brk/dqk,csk/dqk,adtk/bcqk;p)θ(ad,b/d,c/d,ad/bc;p)×(a;rst/q2,p)k(b;r,p)k(c;s,p)k(ad2/bc;t,p)k(dq;q,p)k(adst/bq;st/q,p)k(adrt/cq;rt/q,p)k(bcrs/dq;rs/q,p)kqk=θ(a,b,c,ad2/bc;p)dθ(ad,b/d,c/d,ad/bc;p)×{(arst/q2;rst/q2,p)n(br;r,p)n(cs;s,p)n(ad2t/bc;t,p)n(dq;q,p)n(adst/bq;st/q,p)n(adrt/cq;rt/q,p)n(bcrs/dq;rs/q,p)n(c/ad;rt/q,p)m+1(d/bc;rs/q,p)m+1(1/d;q,p)m+1(b/ad;st/q,p)m+1(1/c;s,p)m+1(bc/ad2;t,p)m+1(1/a;rst/q2,p)m+1(1/b;r,p)m+1}.

    In this section, we consider the case of nontrivial x(k), i.e., x(k)1. For an arbitrary function g(a,b), we will find a function x(k) satisfying (3) and derive a general summation formula.

    Theorem 4.1. Let g(a,b) be an arbitrary function over the complex field C in variables a,b, and ai,bi,ci,pi,qi,ri be arbitrary sequences such that none of the denominators in (13) vanishes. Suppose x(k) is a function over the complex field C which satisfies

    x(k+1)x(k)=g(ak,pk)g(akbkck,pkqkrk)g(akbk,pkqk)g(akck,pkrk)g(bk,qk)g(ck,rk). (12)

    Then for any nonnegative integers m,n,

    mk=nx(k)g(akbk,pkqk)g(akck,pkrk)k1j=1g(bj,qj)g(cj,rj)kj=1g(aj,pj)g(ajbjcj,pjqjrj)=x(n)n1j=1g(bj,qj)g(cj,rj)g(aj,pj)g(ajbjcj,pjqjrj)x(m+1)mj=1g(bj,qj)g(cj,rj)g(aj,pj)g(ajbjcj,pjqjrj). (13)

    Proof. Let

    tk=x(k)g(akbk,pkqk)g(akck,pkrk)k1j=1g(bj,qj)g(cj,rj)kj=1g(aj,pj)g(ajbjcj,pjqjrj).

    Then,

    tk+1tk=x(k+1)g(ak+1bk+1,pk+1qk+1)g(ak+1ck+1,pk+1rk+1)x(k)g(akbk,pkqk)g(akck,pkrk)×g(bk,qk)g(ck,rk)g(ak+1,pk+1)g(ak+1bk+1ck+1,pk+1qk+1rk+1).

    Set

    a(k)=g(bk,qk)g(ck,rk),b(k)=g(ak+1,pk+1)g(ak+1bk+1ck+1,pk+1qk+1rk+1),c(k)=x(k)g(akbk,pkqk)g(akck,pkrk).

    We see that (2) holds. By the assumption (12), we have

    a(k)x(k+1)b(k1)x(k)=c(k).

    Hence the telescoping of tk is given by

    zk=b(k1)x(k)c(k)tk=x(k)k1j=1g(bj,qj)g(cj,rj)g(aj,pj)g(ajbjcj,pjqjrj).

    We get the formula (13) by summing k from n to m and making some simplifications.

    Actually, given an arbitrarily function g(a,b), we can derive an explicit formula for the function x(k) by taking a nonzero initial value x(0). This allows us to construct some new summation identities. In the following part of this section, we will give some applications of Theorem 4.1. We only consider the simple case x(0)=1.

    To derive hypergeometric identities, we take g(a,b)=ab and obtain

    Corollary 4.1. Let ai,bi,ci,pi,qi,ri be arbitrary sequences such that none of the denominators in (14) vanishes. Then for any nonnegative integer m,n, we have

    mk=n(1)k(akbkpkqk)(akckpkrk)k1j=1ajpj(bjqj)(cjrj)kj=1(ajpj)(ajbjcjpjqjrj)=(1)nn1j=1ajpj(bjqj)(cjrj)(ajpj)(ajbjcjpjqjrj)+(1)mmj=1ajpj(bjqj)(cjrj)(ajpj)(ajbjcjpjqjrj). (14)

    Proof. Let g(a,b)=ab. Then by (12) we derive that

    x(k+1)x(k)=akpk.

    Considering the initial value, we have

    x(k)=(1)kk1i=0aipi.

    Hence by Theorem 4.1 we obtain (14).

    As examples, we derive the following hypergeometric and basic hypergeometric identities.

    Example 4.1. Setting n=0 and

    ai=i+1, bi=p+i+1, ci=p2+i+1, pi=1, qi=1, ri=1

    in (14) where 0im, we see that

    x(k)=(1)kk!.

    After some simplifications, we derive that

    mk=0(1)kk(p+k1p)(p2+k1p2)×((k+1)(p+k+1)1)((k+1)(p2+k+1)1)kj=1(j(p+j)(p2+j)1)=(1)m(p+mp)(p2+mp2)(m+1)mj=1(j(p+j)(p2+j)1).

    Example 4.2. Setting n=0 and

    ai=(i+1)2, bi=(p+i+1)2, ci=(p2+i+1)2, pi=1, qi=1, ri=1

    in (14), we see that

    x(k)=(1)k(k!)2.

    After some simplifications, we derive that

    mk=0(1)kk2(p+k1p)(p2+k1p2)(p+k+1p+2)(p2+k+1p2+2)×((k+1)2(p+k+1)21)((k+1)2(p2+k+1)21)kj=1(j2(p+j)2(p2+j)21)=(p+mp)(p2+mp2)(p+m+2p+2)(p2+m+2p2+2)(1)m(m+1)2mj=1(j2(p+j)2(p2+j)21).

    Example 4.3. Let

    x1=cri, x2=bqi, x3=api, x4=dsi

    Setting n=0 and

    ai=x1x2, bi=x1x3, ci=x2x4,
    pi=1/x21/x1, qi=1/x31/x1, ri=1/x41/x2

    in (14), we see that

    x(k)=(b/c;q/r)k(bc;qr)kb2kq2(k2).

    After some simplifications, we derive that [1,Exapmle 6.2]

    mk=0q(k+12)(1abpkqk)(1apk/bqk)(1cdrksk)(1dsk/crk)f(k)=(bc)(11/bc)(ad)(ad1)+q(m2)(bdqmsm1)(bqmdsm)(crmapm)(acpmrm1)bcrmf(m),

    where

    f(k)=(a/c;p/r)k(ac;pr)k(d/b;s/q)k(bd;qs)kbkakp(k+12)(bq/cr;q/r)k(bcqr;qr)k(ds/ap;s/p)k(adps;ps)k.

    Now we consider another choice of the function g(a,b).

    Corollary 4.2. Let ai,bi,ci,pi,qi,ri be arbitrary sequences such that none of the denominators in (15) vanishes. Then for any nonnegative integers m,n, we have

    mk=n(1)k(1akbkpkqk)(1akckpkrk)k1j=1ajpj(1bjqj)(1cjrj)kj=1(1ajpj)(1ajbjcjpjqjrj)=(1)nn1j=1ajpj(1bjqj)(1cjrj)(1ajpj)(1ajbjcjpjqjrj)+(1)mmj=1ajpj(1bjqj)(1cjrj)(1ajpj)(1ajbjcjpjqjrj). (15)

    Proof. Let g(a,b)=1ab. Then by (12) we have

    x(k+1)x(k)=akpk.

    Considering the initial value, we derive that

    x(k)=(1)kk1i=0aipi.

    Hence by Theorem 4.1 we obtain (15).

    Example 4.4. Set n=0 and

    ak=a, bk=b, ck=c, pk=pk, qk=qk, rk=rk

    in (15), we see that

    x(k)=(a)kp(k2).

    After some simplifications, we derive

    mk=0(a)kp(k2)(1abpkqk)(1acpkrk)(b;q)k(c;r)k(ap;p)k(abcpqr;pqr)k=(a1)(abc1)+(a)map(m+12)(b;q)m+1(c;r)m+1(ap;p)m(abcpqr;pqr)m.

    We remark that this formula is a result of Gosper, first published by Bauer and Petkovšek [1,Eq.(6.39)].

    In this section, we will derive an inversion pair via the summation formula (9) and present some applications.

    To construct the inversion pair, we first take n=0 and m=n in (9) and multiply both sides by the common denominator. We thus have

    Lemma 5.1. Let f(a,b) be an arbitrary function over the complex field C in variables a,b which satisfies (8), and ai,bi,ci,di,ei be arbitrary sequences. Then for any nonnegative integers n,

    nk=0bkf(ak,1)f(ak,bkck)f(ak,bkdk)f(ak,bkek)×k1j=0f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)nj=k+1f(bj,1)f(cj,1)f(dj,1)f(ej,1)
    =nj=0f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)nj=0f(bj,1)f(cj,1)f(dj,1)f(ej,1). (16)

    Now we are ready to give the inversion pair.

    Theorem 5.1. Let z and xi,yi (iZ) be indeterminate such that none of the denominators in (18) and (19) vanishes. Suppose f(x,y) is a bivariate function over the complex field C in variables x,y which satisfies (8) and

    f(x,y)=(x/y)f(y,x),f(tx,ty)=f(x,y). (17)

    for any variable t. Set

    fn,k=f(z,z/xkyk)f(z2xk/yk,z)f(zxn,z/yn)f(z2xn/yn,z)nj=k+1f(zxn/yj,1)f(xnyj,1)n1j=kxjf(xn,xj)f(zxn,1/xj) (18)

    and

    gn,k=n1j=kf(xk,1/yj)f(zxk,yj)nj=k+1xjf(xk/xj,1)f(zxkxj,1). (19)

    Then F=(fn,k)n,kZ and G=(gn,k)n,kZ form an inversion pair.

    Proof. Since f(tx,ty)=f(x,y), we have

    fn,n=f(z,z/xnyn)f(z2xn/yn,z)f(zxn,z/yn)f(z2xn/yn,z)=f(z,z/xnyn)f(zxn,z/yn)=1.

    Therefore it holds for l=n.

    Now assume nl+1. Take nnl and

    aj=zxlxn, bj=zxn/yj+l, cj=xnyj+l, dj=xl/xj+l, ej=zxlxj+l

    in (16). The first product of the right hand side of (16) contains the term

    f(anl,enl)=f(zxlxn,zxlxn)

    and the second product of the right hand side of (16) contains the term

    f(d0,1)=f(xl/xl,1).

    Noting that f(x,y)=(x/y)f(y,x) implies that f(x,x)=0, we thus derive that the right hand side of (16) is zero.

    By the relation (17), the left hand side becomes

    nk=lzxnykf(zxlxn,1)f(zxl,zxn)f(z,z/xkyk)f(z,z2xk/yk)×nj=k+1f(zxn/yj,1)f(xnyj,1)f(xl/xj,1)f(zxlxj,1)×k1j=lf(xl,1/yj)f(zxl,yj)f(zxn,1/xj)f(xn,xj)
    =nk=lxnxkf(zxlxn,1)f(zxl,zxn)f(z,z/xkyk)f(z2xk/yk,z)×nj=k+1f(zxn/yj,1)f(xnyj,1)f(xl/xj,1)f(zxlxj,1)×k1j=lf(xl,1/yj)f(zxl,yj)f(zxn,1/xj)f(xn,xj).

    Dividing the factor

    xnf(zxlxn,1)f(zxl,zxn)f(zxn,z/yn)f(z2xn/yn,z)nj=lxj×n1j=lf(zxn,1/xj)f(xn,xj)nj=l+1f(xl/xj,1)f(zxlxj,1),

    we could get it.

    It is straightforward to check that f(x,y)=1x/y satisfies (8) and (17). We thus obtain

    Corollary 5.1. Let

    fn,k=(1xkyk)(1zxk/yk)(1xnyn)(1zxn/yn)nj=k+1(1zxn/yj)(1xnyj)n1j=kxj(1xn/xj)(1zxnxj),

    and

    gn,k=n1j=k(1xkyj)(1zxk/yj)nj=k+1xj(1xk/xj)(1zxkxj).

    Then (fn,k) and (gn,k) form an inversion pair.

    We remark that this inversion pair was first given by Krattenthaler [9,Eq.(1.5)]. From this inversion pair, we can derive several classical inversion pairs.

    Example 5.1. Set z=a, xj=bqj and yj=qj in Corollary 5.1. After some simplifications, we derive the inversion pair

    fn,k=(1aq2k)(b;q)n+k(ba1;q)nk(ba1)k(1a)(aq;q)n+k(q;q)nk

    and

    gn,k=(1bq2k)(a;q)n+k(ab1;q)nk(ab1)k(1b)(bq;q)n+k(q;q)nk.

    This pair is due to Bressoud [3], which was used to study the finite forms of Rogers-Ramanujan identities.

    Example 5.2. Set z=a, xj=bpj and yj=qj in Corollary 5.1. After some simplifications, we derive the following inversion pair

    fn,k=pkn(1bpkqk)(1apkqk/b)(bpnqn;p1)nk(bpnqn/a;p1)nk(1bpnqn)(1apnqn/b)(aq2n1;q1)nk(q1;q1)nk

    and

    gn,k=(bpkqk;q1)nk(bpkqk;q1)nk(aq2k+1;q)nk(q;q)nk.

    Notice that it is an inversion pair that involves rising q-factorials with two different bases, mentioned by Gasper [5]. They used this matrix inverse to derive numerous bibasic, cubic, and quartic summation formulas for basic hypergeometric series.

    By the definition of theta functions, we see that f(x,y)=θ(x/y;p) also satisfies (8) and (17). Therefore,

    Corollary 5.2. Let

    fn,k=θ(xkyk;p)θ(zxk/yk;p)θ(xnyn;p)θ(zxn/yn;p)nj=k+1θ(zxn/yj;p)θ(zxnyj;p)n1j=kxjθ(xn/xj;p)θ(zxnxj;p)

    and

    gn,k=n1j=kθ(xkyj)θ(zxk/yj)nk+1xjθ(xk/xj)θ(zxkxj).

    Then (fn,k) and (gn,k) form an inversion pair.

    This is a pair of inverse matrices due to Warnaar [14] and the elliptic analogue of inverse matrices of Krattenthaler [9,Eq.(1.5)]. Moreover, we can derive the classical pairs of inverse matrices.

    Example 5.3. Set z=ab, xj=aqj and yj=rj in Corollary 5.2, making some simplifications, we can derive a pair of inverse matrices

    fn,k=(1)nkq(nk2)θ(aqkrk;p)θ(qkrk/b;p)θ(aqnrn;p)θ(qnrn/b;p)(aqk+1rn,qk+1rn/b;q,p)nk(r,abrn+k;r,p)nk

    then

    gn,k=(aqkrk,qkrk/b;q,p)nk(r,abr2k+1;r,p))nk

    We note that the case of p=0 corresponds to [5,Eqs. (3.2) and (3.3)] and [9,Eq. (4.3)].

    Example 5.4. Set z=ab, xj=aqj and yj=qrj in Corollary 5.2. After some simplifications, we derive the following inversion pair

    fn,k=(b;q,p)rn(aq;q,p)rnθ(aq(r+1)k;p)θ(bq(r1)k;p)θ(a;p)θ(b;p)×(a,1/b;q,p)k(qr,abqr;qr,p)k(abqrn/b,qrn;qr,p)k(q1rn,aqrn+1;q,p)kqk

    and

    gn,k=θ(abq2rk;p)θ(ab;p)(aqn;q,p)rk(bq1n;q,p)rk(ab,qrn;qr,p)k(qr,abqrn+r;qr,p)kqrk.

    It is due to Warnaar [14,Eq.(3.5)], who used it to set up a series of summation and transformation formulas for terminating, balanced, very-well poised, elliptic hypergeometric series.

    The work was supported by the National Natural Science Foundation of China (grant 11771330). The authors would like to thank the referees for their careful reading of the manuscript and for their constructive suggestions provided in the reports.



    [1] Multibasic and mixed hypergeometric Gosper-type algorithms. J. Symbolic Comput. (1999) 28: 711-736.
    [2] Generalized bibasic hypergeometric series and their $\begin{document}U\end{document}(n)$ extensions. Adv. Math. (1997) 131: 188-252.
    [3] A matrix inverse. Proc. Amer. Math. Soc. (1983) 88: 446-448.
    [4] An extension of Zeilberger's fast algorithm to general holonomic functions. Discrete Math. (2000) 217: 115-134.
    [5] Summation, transformation, and expansion formulas for bibasic series. Trans. Amer. Math. Soc. (1989) 312: 257-277.
    [6] G. Gasper and M. Rahman, Basic Hypergeometric Series, Second Ed., Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511526251
    [7] Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA (1978) 75: 40-42.
    [8] Summation in finite terms. J. Assoc. Comput. Mach. (1981) 28: 305-350.
    [9] A new matrix inverse. Proc. Amer. Math. Soc. (1996) 124: 47-59.
    [10] The (f,g)-inversion formula and its applications: The (f,g)-summation formula. Adv. in Appl. Math. (2007) 38: 227-257.
    [11] P. Paule and C. Schneider, Towards a symbolic summation theory for unspecified sequences, In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, 351–390, Texts Monogr. Symbol. Comput., Springer, Cham, 2019.
    [12] M. Petkovšek, H. S. Wilf and D. Zeilberger, A=B, A K Peters, Ltd., Wellesley, MA, 1996.
    [13] C. Schneider, Symbolic Summation in Difference Fields, Ph. D. thesis, J. Kepler University, 2001.
    [14] Summation and transformation formulas for elliptic hypergeometric series. Constr. Approx. (2002) 18: 479-502.
    [15] The method of creative telescoping. J. Symbolic Comput. (1991) 11: 195-204.
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